Landmark XIV-I Uniform internal retarded amplitudes at the diagonal–flow-out transition Exact reconstruction of both K 5 placements and the remaining common-composition gate

We continue the operator route at XIV-H's missing internal amplitude theorem. The frozen pair, source, inverse coefficient, switches, width, and partial-diagonal ambiguity class are unchanged. The residual metric variable is retained.

For the two internal placements of $K_5$, we define a finite distribution-valued Fourier-amplitude module. It is not a classical symbol space on one ordinary Lagrangian. The fixed massless retarded multiplier has an explicit simple-pole decomposition near either nonzero characteristic sheet. This decomposition relates the noncharacteristic multiplier to the separated light-cone kernel, with its retarded normalization fixed by the already existing distribution.

We prove uniform annular estimates across the pole, continuity of the reconstruction map, and an exact finite decomposition into the highest normal-frequency degree and its lower-degree remainder. Both placements are reconstructed in their actual coordinates, with uniform control on compact sets of all untested variables.

This constructs the named internal amplitude map at I2. It does not construct the joint characteristic multiplication required for the grouped six-branch coefficient. A common precomposition presentation is supplied, but the operator map composing it with the grouped cross-lines on the retained characteristic locus is missing. The first unresolved gate is therefore I3. No joint section, characteristic prescription, transverse degree, gluing theorem, or output admission is released. In particular, no ${R_1}^{(1)}$ is admitted in ${\mathcal{E}}_H$, and XV-B remains closed.

Figure fig:split-XIV-I records one view of the same separation. Figure fig:chain-XIV-I records one view of the same separation. Figure fig:pair-XIV-I records one view of the same separation. Figure fig:gate-XIV-I records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-XIV-I records that separation. The drawing adds no hypothesis.

and distance to XV-B{ Constructed at I2: an explicit internal amplitude space, its fixed pole normalization, uniform distributional frequency estimates, exact reconstruction, and complete finite lower-degree remainder for both $K_5$ placements.

First missing gate: I3. The construction is internal and precompositional. No justified joint characteristic operator map for ${\mathcal{T}}_{\mathrm{form}}$ is supplied.

Not released: a joint principal-symbol section, a characteristic prescription, J3–J5, or an admitted ${R_1}^{(1)}$.

XV-B remains closed. XV-A's free carrier is an exit prerequisite, not an analytic substitute for the missing coefficient.}

All manuscript handoffs below are premises with their predecessor domains. No incidence ledger, elementary phase theorem, or clean-excess theorem is re-proved.

A successor receives:

1. the unchanged XIV-G package;

2. the untested source incidence and its regular relation;

3. the exact coordinate-leg convolution map and its regular conormal representation;

4. the separate complete scalar-leg root map;

5. the exact two-placement $K_5$ tensor pullbacks;

6. their regular relation charts and orders;

7. the clean diagonal/flow-out intersection and the exclusion of a single ordinary Lagrangian identification;

8. the precise missing uniform internal amplitude theorem and the H2 stop.

It receives no transverse degree of the target, common characteristic representation, cancellation theorem, or permission to call the tangent defect an excess. It receives no ${R_1}^{(1)}$ XIVH.

Define constant fiber maps by

where $I$ indexes symmetric metric components. Let $R$ be the kernel of ${E_0}^{\mathrm{R}}$.

The full differentiated source has the finite local form

with every coefficient taken from the actual density-weighted source. Before residual testing its kernel is

This is a defined regular incidence supported on $v=a=z$.

Writing the response kernel with all three species indices,

the exact coordinate-leg map on the retained patch is

The complete scalar-leg root, with $F={E_m}^{\mathrm{A}}f$, gives

The mass term is retained.

On the separated chronological patch, both kernels have regular conormal representations of upper order $3/2$ on

The $y$ covector is zero XIVH.

The explicit output index $L$ in (eq:T) only removes an index suppression in the predecessor. It changes no map.

In the normalization $\square {E_0}^{\mathrm{R}}=\mathrm{id}$,

The second equality is an identity of existing internal retarded kernels, not a cross-line product XIVH.

Put

The complete untested mixed kernel of $B_5$ is

\begin{split}
 K_5^{I,J}(a,b,z)
 =\partial_{b^\mu}\Bigl[
 &f(b)\sigma_\varepsilon(a)
 \partial_{a^\alpha}R(b-a)
 \mathsf A^{\alpha\beta,I}\partial_{a^\beta}L^{\mu,J}(a,z)\\
 +{}&
 f(b)\sigma_\varepsilon(z)
 \partial_{z^\alpha}R(b-z)
 \mathsf A^{\alpha\beta,J}\partial_{z^\beta}L^{\mu,I}(z,a)
 \Bigr].
 \label{eq:K5}

The two underlying products are

They are defined tensor-product pullbacks, including at $a=z$.

Their exact finite differential formulas have total kernel derivative order at most four. On regular separated-cone charts their upper conormal order is one. On comparable-frequency, noncharacteristic internal diagonal charts their upper order is $1/2$.

For the first placement, near a nonzero internal null covector, the diagonal piece and limiting flow-out piece are distinct sixteen-dimensional Lagrangians. They intersect cleanly in dimension fifteen and have tangent sum of dimension seventeen. No single ordinary sixteen-dimensional Lagrangian contains both full local pieces there. The second placement has the corresponding reversed internal separation.

The first missing theorem was a uniform internal amplitude description through this transition:

No common characteristic representation was released XIVH.

The clean intersection is not called non-clean. Its tangent-sum count is not XIV-F's linearized matching count.

analytic boundaries

XIV-G's tested source is a local differential operator of order at most two; its local polynomial symbol is not either response symbol. Smooth residual testing can hide attachment-diagonal wavefront directions. Its balanced-kernel counterexample is not a model of the characteristic singularity of $K_{\mathrm{ret}}$.

The ordinary final product of its two individually defined numerical factors fails the sufficient product criterion at the retained covectors. That audit supplies no full-pair nonvanishing theorem XIVG.

XIV-F's rank-two object is a tangent defect. Its excluded matching has actual dimension fifteen and linearized dimension seventeen. The covector-sum map has rank fourteen, one-dimensional fibers, and an isotropic, non-Lagrangian image in $T^{\ast }W$. Its regular-stage formula is

These are boundaries, not relations used in the present amplitude construction XIVF.

For smooth directions,

where

The complete scalar root is

These are exact precomposition identities, not characteristic products. The tested attachment bound is not a uniform untested composition theorem XIVE.

On the retained patch,

Both gravitational-response legs are retained. The numerator $Q\lambda$ belongs only to the leading differential fiber calculation for $B_6$ XIVD.

The common space is

with the residual variable $z$ retained. The characteristic test locus is

It has codimension eight in $W$.

The three separated cone conormals have the copied positive relation. Defined interaction collision contacts vanish on the retained collision-free attachment neighborhood XIVC,XIVD.

XIII-C's four obligations remain:

1. a characteristic prescription or complete-pair cancellation;

2. uniform transverse estimates at three- and four-vertex collision strata and their intersections;

3. compatible partial contacts with singular spectators;

4. output wavefront control after proper pushforward.

The parallel temporal kinetic primitive has collision scaling degree eight in codimension four XIIIC.

XIII-B supplies the admitted tree datum and the convention

The strictly ordered full-support values are zero or the scaled commutator. Properness does not define an absent distributional composition XIIIB.

The inverse, slots, and roots retain their normalization:

The full observable contains the scalar-sourced slot once XIIE.

The actual source is

in component notation, with the fixed unintegrated cubic representative of XII-C. No source coefficient is deleted XIIC.

-A is only an exit prerequisite

XV-A states a free-carrier release $({{\mathcal{E}}_{\mathcal{H}}}^{\mathrm{BV}},{\hat{s}}_0)$ and a generator-preserving embedding obstruction for the relative classical complex XVA. Neither is used to prove an amplitude estimate or define a cross-line product here. Their stated free-carrier conclusion is recorded only when discussing the exit to XV-B.

XIV-A and XIV-B are not analytic inputs. The value $-8I$ is not a normalization. XIII-E and a resolved paired construction remain unentered.

1: the frozen route and the amplitude class

Use

All inherited functions and the width $\epsilon >0$ remain fixed. The pair is

and

The target remains

The difference stays grouped. No mixed $H$–$G_{\mathrm{act}}$ product is declared.

The selected class consists of internal representations on the relations already named in XIV-H, with both $K_5$ placements and $z$ retained. It permits:

1. partial Fourier transforms of the already defined internal kernels;

2. explicitly defined distribution-valued normal amplitudes, with a proved reconstruction map, uniform estimates, and a stated remainder;

3. regular conormal charts and named pullbacks or operator maps whose hypotheses are checked.

It does not require the union of the internal diagonal and flow-out pieces to be one ordinary Lagrangian.

It does not permit an unproved generalized composition with the quantum cross-lines. Such a composition is the later I3 obligation. No common product phase, resolved base space, frequency-regulator family, or characteristic prescription for a new product is introduced.

The construction below is a finite, explicit amplitude module. It is not an assertion of a general paired-distribution calculus.

The endings are applied to the object at the gate reached:

1. Constructed: that object has its stated domain, amplitude or degree, and wavefront control.

2. Proved obstructed: an incompatibility excludes every candidate in Definition def:class.

3. Unresolved: a necessary amplitude, common composition, section, prescription, or J-gate estimate is missing.

The earlier exclusion of a single ordinary Lagrangian is not a universal obstruction to this class.

Only admission of ${R_1}^{(1)}$ in ${\mathcal{E}}_H$ at I6 could open XV-B. An I2 amplitude theorem opens I3 only.

2a: the fixed retarded multiplier and its transition pole

convention

For a relative variable $r=(t,\mathbf{x})$, use

Inverse Fourier transformation has the factor $(2\pi {)}^{-4}$. Thus ${{\partial }_r}^{\beta }$ has multiplier $(i\kappa {)}^{\beta }$, where

The $\kappa$ in this section is a frequency variable, not the interaction coupling.

All Fourier calculations concern the already fixed distribution $R$. They choose no new characteristic value for (eq:target).

The tempered Fourier transform of $R$ is

On $\rho >0$,

It is homogeneous of degree $-2$ in the distributional sense.

Proof. The spatial Fourier transform of (eq:R) is

with continuous value $\Theta (t)t$ at $\rho =0$. Multiplying this existing distribution by $e^{-ϵt}$ on its support and taking its temporal Fourier transform gives

The damped distributions converge to $R$ in ${\mathcal{S}}^{\prime }({\mathbb{R}}^4)$, as follows directly from its cone formula and the rapid decay of Schwartz tests. Fourier continuity gives (eq:MR).

Partial fractions followed by the same distributional limit give (eq:partialfraction) when $\rho >0$.

The cone formula has physical homogeneity degree $-2$ in dimension four. Its Fourier transform consequently has degree $-2$ as well. Alternatively the same identity follows by positive dilation in (eq:MR).

The damping in this proof computes the Fourier transform of an existing kernel. It is not a regulator for a graph product, and no restoring limit of the quantum coefficient is used.

uniform pole chart on either characteristic sheet

For $s\in \{+1,-1\}$, define

Take a conic neighborhood of that nonzero sheet satisfying

The opposite factor $\omega +s\rho$ is then nonzero.

Let

On the stated conic neighborhood,

The coefficient $a_s$ is smooth and homogeneous of degree $-1$, and $b_s$ is smooth and homogeneous of degree $-2$ there.

Off ${\lambda }_s=0$, the complete expression equals $-1/p(\kappa )$. Its singular and regular pieces in (eq:pole) are both retained.

Temporal inverse Fourier transformation gives exactly

Thus the pole and noncharacteristic multiplier reconstruct the same retarded kernel, whose separated-cone coefficient is the one in (eq:R).

Proof. For $s=+1$, isolate the first term of (eq:partialfraction); for $s=-1$, isolate the second. The opposite denominator is smooth on the selected chart, giving (eq:pole). The homogeneities are immediate.

Away from the pole, both boundary values become their ordinary reciprocals. Combining the two fractions gives $-1/p$.

The elementary Fourier identity

follows by Fourier transforming the right-hand side with positive damping. Applying it to (eq:partialfraction) gives $\Theta (t)\sin (\rho t)/\rho$. Its spatial inverse transform is the fixed cone kernel (eq:R), by Lemma lem:multiplier and Fourier injectivity.

The $i\pi \delta ({\lambda }_s)$ part is fixed by the original retarded support. It is not an adjustable internal counterterm. Replacing it would change $R$ and hence the frozen pair.

This pole chart is not a characteristic prescription for ${\mathcal{T}}_{\mathrm{form}}$. Only one pre-existing internal line is being represented.

2b: a finite distribution-valued amplitude space

coordinates for the two placements

For the first placement use

For the second use

The copied maps

are diffeomorphisms onto their images.

On either patch $d_{\pm }$ is nonzero and future directed. Near its null cone choose smooth coordinates

for $d_{\pm }$. The derivative of ${d_{\pm }}^2$ is nonzero there, so these are ordinary regular coordinates.

Let $u_{+}=a$ and $u_{-}=z$. An adapted chart is

Its dimension is sixteen, and $r_{\pm }$ retains the residual metric incidence rather than testing it away.

The construction uses separate normal Fourier transformations of already defined tensor factors. It assigns no common product phase to the six-branch quantum coefficient.

amplitude module

Let $\theta$ be the outer normal frequency dual to $q$. For $j\ge 0$ and a four-dimensional multiindex $\beta$, set

For an adapted chart $V$ and an integer $m\ge -2$, let ${{\mathfrak{A}}_{\mathrm{R}}}^m(V)$ consist of finite expressions

with the finite inherited tensor components understood.

The coefficients carry their usual compact-set smooth seminorms. The frequency factors are distributions, not ordinary smooth symbols at $p(\kappa )=0$.

The module is filtered by the polynomial degree $j+|\beta |$. Its reconstruction is defined by

Normal inverse Fourier transformation is taken on each tensor factor, and coefficient multiplication is performed afterward.

This is a specific finite module with fixed generators. It is not the unrestricted space of tempered amplitudes. No ordinary principal-symbol bundle over a single Lagrangian is asserted.

The polynomial coefficient presentation is unambiguous as a frequency amplitude: multiplication by $M_{\mathrm{R}}$ annihilates no nonzero polynomial, since $M_{\mathrm{R}}=-1/p$ on a nonempty open set. The reconstruction need not be injective, because physical coefficient identities such as $q\delta (q)=0$ may occur. Injectivity of reconstruction is not required.

estimates through the pole

For a distribution $A$ in five frequency variables, $A(L\cdot )$ denotes pullback by the dilation $(\theta ,\kappa )\mapsto (L\theta ,L\kappa )$.

For every compact interval $J$, there are a slightly larger compact interval $J^{\prime }$ and a constant $C_J$ such that

The same type of finite-order estimate holds after any fixed number of derivatives of $B_{-}$.

Proof. Choose a smooth even function equal to one near zero. In the principal-value integral subtract $\psi (0)$ times that function. Near zero the remaining numerator is bounded by $|\lambda |$ times a first-derivative bound. Away from zero the denominator is bounded away from zero on the remaining compact integration set. The delta part is bounded by $|\psi (0)|$. This gives the estimate.

For a derivative, transfer that derivative to the test function and apply the same argument.

Let $K$ be a compact subset of a normalized frequency annulus in $(\theta ,\kappa )$ contained in one of the pole charts of Proposition prop:pole. The compact set may meet ${\lambda }_s=0$.

If $d=j+|\beta |$, then

as distributions. For every fixed frequency multiindex $\nu$, there are $N$ and $C$ such that

For amplitudes (eq:amodule), these estimates are uniform after any fixed number of coefficient derivatives on a compact subset of $V$. No separation from the pole is required.

Proof. Lemma lem:multiplier gives degree $-2$ for $M_{\mathrm{R}}$. The polynomial has degree $d$, giving (eq:homogeneity). A distributional frequency derivative lowers the homogeneity by one.

It remains to verify finite seminorm control on $K$, including its intersection with the pole. In the coordinates $({\lambda }_s,\xi )$, (eq:pole) is a smooth homogeneous coefficient times $B_{-}({\lambda }_s)$ plus a smooth term. The change of frequency coordinates is regular where $\rho >0$. On $K$ all its coefficients and the required finitely many derivatives are bounded.

Apply Lemma lem:PVbound in ${\lambda }_s$, integrate over the remaining compact variables, and include the polynomial factors and any fixed derivatives. This gives a finite-order distribution bound. The homogeneity then makes its constant independent of $L$.

For parameter-dependent coefficients, the same argument is multiplied by their finite smooth seminorms on a compact subset of $V$.

These are estimates for internal frequency distributions. They are not normal rescaling estimates of ${\mathcal{T}}_{\mathrm{form}}$ at $N$.

finite remainder

For an amplitude of degree at most four, split its coefficient polynomial by degree:

Terms with no degree-four contribution have $A_{[4]}=0$.

The splitting (eq:split) is exact. On normalized annular tests,

is uniformly bounded, and

in the finite-seminorm distribution topology of Theorem thm:dyadic, uniformly on compact parameter sets.

Every lower-degree term of the finite differential formula is retained in $A_{[\le 3]}$. The regular opposite-sheet term $b_s$ in (eq:pole) remains part of the full multiplier in both pieces.

Proof. Each degree-$d$ term scales as $L^{d-2}$. For $d\le 3$, division by $L^2$ gives at most $L^{-1}$. There are finitely many terms, so the uniform estimates can be summed. No term of $M_{\mathrm{R}}$ was dropped in defining the generators.

This is not a smoothing remainder theorem for the quantum coefficient. It is a proved, complete lower-degree estimate for the named internal module.

reconstruction

The map (eq:reconstruction) is a continuous linear map from the coefficient module, with its compact-set smooth topology, to ${\mathcal{D}}^{\prime }(V)$.

It is also the iterated normal inverse Fourier transform of the generators (eq:generators), followed by the specified coefficient multiplication.

On compact subsets of the original $W$ chart, its estimates hold for arbitrary joint tests in $(a,b,z,y)$. No restriction to a fixed singular residual value is used. A fixed derivative in the original variables raises the finite polynomial degree by at most its differential order.

Proof. The tensor product ${\delta }^{(j)}(q)\otimes {{\partial }_r}^{\beta }R(r)$ is defined. Its pullback to the adapted chart merely adds passive variables. Multiplication by a smooth coefficient is defined.

For a compactly supported joint test, the pairing is first a finite $q$ derivative evaluated at $q=0$, then a pairing with the fixed distribution ${{\partial }_r}^{\beta }R$, and integration over the remaining compact tangential variables. The distributional finite-order estimate for $R$ on the projected compact set bounds this by finitely many derivatives of the coefficient and test. This proves continuity.

Separate Fourier inversion gives

This proves the reconstruction identity without evaluating an oscillatory amplitude at a singular point.

The adapted coordinate maps are diffeomorphisms. Pulling the estimates back therefore bounds arbitrary joint tests in the original variables, including $z$. The chain rule expresses an original-variable derivative as smooth-coefficient derivatives in the adapted coordinates. Its normal derivatives add polynomial factors and its tangential derivatives differentiate coefficients. This proves the final statement.

Uniformity in $z$ means this joint distributional control. It does not mean that the kernel admits a pointwise smooth slice at every $z$.

{I2c: application to both $K_5$ placements}

coefficient extraction from the exact formula

In either adapted chart, the outer line is

The inner line remains $R(r_{\pm })$, including its vertex.

The chain rule and the Leibniz rule applied to the complete formula (eq:K5) give coefficient arrays ${c_{\pm ,j\beta }}^{I,J}$ such that

Factors $1/(2\pi )$ and all coordinate density factors are included in the coefficients.

Equation (eq:coefficientform) is an exact coefficient extraction rule: expand the finite differential operator in (eq:K5) in the adapted coordinates, transfer its normal derivatives to the two indicated factors, and retain every derivative of every coefficient. It is not a tensor truncation.

Define

The amplitudes (eq:A5) reconstruct the two placements in (eq:K5) exactly:

after the two adapted coordinate maps are pulled back to $W$.

They satisfy the uniform transition estimates and exact lower-degree remainder of the preceding section. Both metric-slot placements and every inherited coefficient are retained.

Proof. The copied formula differentiates the two retarded factors at most four times. In adapted coordinates, its tangential derivatives act on smooth coefficients, whereas its normal derivatives act on $\delta (q_{\pm })$ or $R(r_{\pm })$. The finite Leibniz expansion is therefore precisely (eq:coefficientform).

Theorem thm:reconstruction proves the equality. Theorem thm:dyadic and Corollary cor:remainder apply to its finite coefficient arrays. No metric slot has been tested against $G$ or assigned a value.

The amplitudes use the two original relative variables $r_{+}=a-z$ and $r_{-}=z-a$. The internal time orientation is therefore handled separately for each placement; one is not substituted for the other.

to the predecessor's regular charts

The same internal module gives:

1. the ordinary noncharacteristic chart by replacing the full multiplier with its actual value $-1/p$ on an open set avoiding $p=0$;

2. the separated-cone chart by inverse Fourier reconstruction of the full pole expression, giving the exact retarded cone distribution;

3. the transition itself by the distributional expression (eq:pole), with the uniform bounds proved through the pole.

The copied regular conormal orders remain upper orders in their own charts. The distributional degree $2$ of the degree-four amplitude in (eq:A5) is not a replacement for either order.

In particular, there is no contradiction between a regular diagonal upper order $1/2$, a separated two-cone upper order one, and the internal distributional dilation estimate. They describe different symbol presentations and different normal variables.

joint internal wavefront bound

Let ${{\mathcal{C}}_{\mathrm{R}}}^0$ consist of:

together with

and the nonzero forward-cone directions

Here $r^{♭}=\eta (r,\cdot )$. This is the usual retarded-kernel wavefront upper bound, with zero covectors on its support adjoined.

In either adapted chart, define

The covectors are pulled back to $T^{\ast }W$ by the actual adapted coordinate maps.

The reconstructed kernels satisfy

The bounds retain the internal diagonal and the separated flow-out, as well as the directions where one tensor factor has zero covector.

They are uniform bounds on the untested kernel, not merely bounds after smooth residual testing.

Proof. At a nonzero point of the retarded cone, (eq:R) is delta-type on its smooth defining hypersurface, so its wavefront is conormal there. At the vertex the asserted upper bound includes every nonzero covector and is therefore valid.

The tensor-product wavefront bound, with zero covectors adjoined on supports, gives (eq:Gamma5) for the two independent factors in (eq:coefficientform). The coordinate pullbacks are diffeomorphisms. Differentiation and smooth multiplication do not enlarge these bounds. The finite sum gives the result Hormander (Section 8.2).

The $b$ covector in the outer cone component is not set to zero. The balanced-kernel testing example of XIV-G is not used to model these directions.

In the finite retarded-pole module of Definition def:amodule, the I2 amplitude map for both $K_5$ placements is Constructed:

1. one explicit module contains their regular diagonal, separated flow-out, and internal transition presentations;

2. (eq:pole) fixes the relation between the noncharacteristic multiplier and the retarded cone amplitude;

3. the untested reconstruction has uniform compact-set control with $z$ retained;

4. every finite lower-degree term is included in the exact remainder (eq:split);

5. the reconstructed internal kernels have the wavefront bounds (eq:Gamma5).

This is not a single-Lagrangian principal-symbol theorem. It supplies no joint characteristic product with the quantum cross-lines. It opens I3 only.

Proof. Proposition prop:pole proves the transition identity. Theorems thm:dyadic and thm:reconstruction prove the uniform estimates and reconstruction. Corollary cor:remainder proves the complete finite lower-degree statement. Proposition prop:both applies these results to both exact placements, and Proposition prop:WF supplies the untested wavefront control.

3: the common precomposition sum and the missing joint map

I2 is now supplied in its named amplitude module. I3 is therefore entered, but its target is stronger than internal reconstruction.

other precomposition pieces

The regrouped $B_4+B_7$ kernel can be written directly with the existing coordinate-response line $L$:

This retains the complete $C_G$ term. It does not reintroduce the canceled explicit ${\partial }^2\psi$ term.

Let ${\mathsf{D}}^{\alpha \mu ,I,J}$ be the constant bilinear fiber map determined by

The local $B_6$ piece has kernel

On the retained patch,

has a defined finite operator presentation with $z$ retained.

The pieces have the following internal descriptions:

1. $K_{47}$ is a finite derivative of the two independent separated retarded factors $R(b-a)R(b-z)$;

2. $K_5$ has the two constructed internal amplitude maps of Theorem thm:I2;

3. $K_6,K_8,K_9$ have finite conormal amplitudes on ${\Lambda }_{\mathrm{resp}}$.

On regular conormal charts their upper orders are, respectively, one for $K_{47}$, $1/2$ for $K_6$, and $3/2$ for $K_8,K_9$. These are internal orders only.

Proof. Formula (eq:K47) is obtained by inserting the kernel of $X$ into the copied coefficient $C_G$ and transferring the scalar derivative to the resulting kernel. Its two relative variables are independent under the diffeomorphism

Both endpoints are separated on this patch. The two cone differentials are independent, and the total derivative count is at most four. In dimension sixteen and codimension two the copied conormal convention gives upper order one.

For (eq:K6), transfer the divergence to the incoming retarded line and then transfer the scalar derivative. The two signs cancel. The actual bilinear local vertex forces $a=z$. Its two normal derivatives give upper order $2-(16-10)/4=1/2$.

The $K_5$ statement is Theorem thm:I2; the two response statements are Input statement input:Hresponse. The copied regrouping then gives (eq:presum).

This proposition constructs a common precomposition presentation of the observable Hessian. It does not construct the operator map required for (eq:target) at $N$.

grouped cross-line distribution is not the joint product

The two-line tensor

is a defined distribution in its two independent relative variables. Its pullback by

is defined, since this map is a diffeomorphism onto its image. Denote the resulting grouped distribution by ${\mathcal{C}}_{\mathrm{cross}}$.

Only this grouped pullback is used. It is not expanded into mixed $H$–${G_{\mathrm{act}}}^{\mathrm{A}}$ products, and it is not replaced by the numerical factor of XIV-G's audit.

The local mixed Hessian of ${\mathcal{V}}_g$ supplies its inherited finite fiber contraction and differential operators. Applying those operators to a defined distribution is legitimate. It does not define multiplication of that distribution by $K_{\mathrm{ret}}$ at a characteristic covector cancellation.

actual missing I3 map

Let ${\mathcal{K}}_{\mathrm{int}}$ denote the finite direct-sum coefficient spaces in Proposition prop:precomposition. Their reconstruction map is defined on the full retained patch.

A successful I3 construction must supply a map of the form

whose value on the reconstructed six-branch sum is the specified contraction with ${\mathcal{C}}_{\mathrm{cross}}$ and the mixed vertex Hessian, and which:

1. agrees with the canonical composition on its proved domain;

2. is defined on tests meeting $N$;

3. retains the prescribed grouped cross-lines and all tensor coefficients;

4. has a proved complete-graph wavefront bound appropriate to that composition.

The internal Fourier-amplitude module proves none of these requirements for multiplication by ${\mathcal{C}}_{\mathrm{cross}}$. Its estimates are estimates on the internal amplitude before this new operation.

On the inherited microlocally admissible domain $U_{\mathrm{can}}$, the reconstructed sum (eq:presum), composed with ${\mathcal{C}}_{\mathrm{cross}}$ and the specified vertex operators, equals the canonical grouped expression (eq:target).

No equality

follows from this representation.

Proof. Each reconstruction is an equality of already defined tree distributions. Restrict those equalities to the open domain where the subsequent products and pullbacks have been proved. The ordinary distributional operations there respect equality, giving the first assertion.

The internal reconstruction proves no new normal-set condition for that subsequent product, so it cannot identify a larger domain.

The copied opposite-covector audit prevents declaring the ordinary final product theorem automatically applicable at $N$. It is not retried here.

A complete-pair cancellation or a generalized composition could in principle provide (eq:jointmap). Neither is established by the exact internal pole decomposition.

I2 is Constructed for the named internal amplitude map. I3 has a defined common precomposition presentation and a canonical-domain representation.

The joint characteristic operator map (eq:jointmap) on tests meeting $N$ is not constructed. No complete-pair cancellation theorem or alternative composition theorem is supplied.

The ending for the grouped characteristic target is therefore Unresolved at I3, within G2/F2/E1/D2/J2. No incompatibility excluding every representation in Definition def:class is proved.

Proof. Theorem thm:I2 discharges the stated internal amplitude gate. Proposition prop:precomposition supplies the exact internal sum, and Proposition prop:canonical supplies its composition only on the inherited admissible domain.

Those maps do not act on a test crossing $N$ by the required characteristic composition. The uniform frequency estimates control the internal retarded pole, not its product with the grouped cross-lines. No theorem defining that latter operation has been proved. This is precisely the missing I3 requirement.

Nothing in the internal construction proves that all possible such maps fail. The ending is accordingly Unresolved, not Proved obstructed.

distinction at I3{ The internal amplitude is no longer the missing H2 object. Its pole, reconstruction, uniform estimates, and finite remainder have been supplied.

The missing object is now the joint characteristic operator map for the complete grouped coefficient. Calling the precomposition sum that map would skip a different product hypothesis.

No quantum coefficient is released by moving the first missing gate from H2 to I3.}

4: joint section or prescription, not entered

Because I3 is Unresolved, I4 is not entered as a construction.

If I3 were later supplied on the same route, the selected continuation would be the joint-section and complete-remainder exit. No characteristic prescription is selected here.

That later theorem must control:

1. lower terms of every numerical line, including the massive cross-line contributions;

2. every internal vertex and its actual operator map;

3. attachment-diagonal derivatives;

4. the two geometric defect directions without treating them as integration fibers;

5. the complete actual source;

6. the derivative and mass terms of the scalar root;

7. the complete grouped cross-line difference;

8. tensor and branch summation on one justified common representation.

The finite internal remainder of Corollary cor:remainder is not that theorem. Its frequency degree is not a transverse degree of ${\mathcal{T}}_{\mathrm{form}}$.

No zero joint section is assumed, and no nonzero joint section is inferred from the $B_6$ numerator $Q\lambda$.

The results of XIV-I prove neither cancellation nor failure of cancellation of ${\mathcal{T}}_{\mathrm{form}}$.

Proof. The new amplitude reconstruction concerns $K_5$ before cross-line composition. The other new formulas identify the common internal sum. Neither constructs (eq:jointmap) at $N$ or assigns a joint principal-symbol section there.

No complete remainder for that absent joint operation is supplied. The internal homogeneous-degree splitting cannot replace it.

unentered prescription interface

The alternative prescription route remains a different construction. It would have to name all seven data:

1. ambient space;

2. reference distribution on a proved domain;

3. exact extension locus;

4. agreement with the grouped frozen prescription;

5. uniform transverse degree;

6. a closed wavefront bound retaining off-locus singularities;

7. normalization or a uniqueness theorem.

The present locus is still the codimension-eight $N={\tilde{\mathcal{N}}}_{\mathrm{att}}\subset W$. It is not converted into a codimension-four numerical problem.

No Brunetti–Fredenhagen or Dang extension theorem is applied to the missing target BF,Dang. XIII-E and a resolved paired construction stay unentered. There is no new contact class.

5: J3–J5 remain unopened

J3 requires a distribution supplied by I4 before uniform transverse estimates can be assigned to it. No such distribution is supplied.

The parallel-line scaling degree eight remains a collision degree. Neither that value, the copied internal conormal orders, nor the distributional frequency degree in Theorem thm:dyadic is used as a transverse degree of ${\mathcal{T}}_{\mathrm{form}}$.

J4 would require composition with the singular spectators and permitted partial contacts, together with overlap agreement and the inherited normalizations. No joint gluing is performed.

J5 would require the complete output wavefront theorem after the actual proper pushforward. The residual-field target is a compact microcausal degree-two kernel, for example with the sufficient bound

under the fixed flat cotangent identifications.

This residual-field output bound is different from an intermediate attachment-kernel bound. Properness of the existing tree datum cannot supply it.

The internal wavefront bound of Proposition prop:WF is not the J5 bound: its attachment variables have not been composed and pushed forward in the required quantum graph.

6: admission and the XV-B exit

An admission line would have to identify ${R_1}^{(1)}$ as the pushed-forward coefficient of the constructed, normalized joint distribution and prove

with XIII-B's retarded convention.

No such line is asserted in this paper. The tree coefficient is not substituted for ${R_1}^{(1)}$.

Under the stated series contract, XIV-I does not open XV-B.

Proof. XV-B requires both the stated free carrier of XV-A and an admitted ${R_1}^{(1)}$ in ${\mathcal{E}}_H$. The latter has not been supplied: I3 is Unresolved and I4–I6 have not been discharged. The conjunction of exit prerequisites therefore fails.

If a later paper supplied the admission line, XV-B could attempt one relative interacting differential through one explicit order in $\kappa$ and $\hslash$ on the free carrier. The relative classical complex would remain a comparison object, not an identified quantum subalgebra.

XV-A's stated odd-sector antifield coupling

belongs to that carrier prerequisite. It does not normalize any amplitude in this paper.

A quantum master-equation defect, anomaly classification, or promotion of ${\mathcal{C}}_{\xi }$ would still belong to subsequent gates, not automatically to XV-B.

No ${\hat{s}}_0$, $s_{\mathrm{ext}}$, $s_{\Psi }$, or $s_0$ is applied to the absent coefficient. No Ward defect is formed.

and exact release

For the unchanged frozen pair and the route fixed in I1:

1. the fixed internal retarded multiplier has the exact pole decomposition (eq:pole);

2. the finite amplitude module has the uniform estimates, continuous reconstruction, and exact lower-degree remainder proved in Section sec:amplitudespace;

3. both $K_5$ placements have the amplitude maps (eq:A5), reconstructing their complete untested kernels with the stated wavefront bounds;

4. the retained six-branch tree kernel has the defined precomposition presentation (eq:presum);

5. no common characteristic operator map for the grouped target on tests meeting $N$ is constructed;

6. no ${R_1}^{(1)}$ is admitted in ${\mathcal{E}}_H$.

The named internal amplitude ending is Constructed at I2. The grouped characteristic coefficient remains Unresolved at I3, within G2/F2/E1/D2/J2. XV-B remains closed.

No universal obstruction to the class fixed in I1, full cancellation theorem, or admitted quantum output is asserted.

Proof. Lemma lem:multiplier and Proposition prop:pole prove (i). Theorems thm:dyadic and thm:reconstruction, with Corollary cor:remainder, prove (ii). Propositions prop:both and prop:WF prove (iii), summarized in Theorem thm:I2.

Proposition prop:precomposition proves (iv). Theorem thm:I3 identifies (v). Sections sec:I4–sec:I6 record the undischarged subsequent gates and establish (vi). Corollary cor:XVB gives the exit consequence.

What has advanced

The internal amplitude is now more than the assertion that a tree kernel exists. Its frequency singularity is explicitly a fixed retarded simple pole plus its regular companion. Its estimates hold on tests crossing that pole, and its inverse transform reconstructs both the diagonal chart and the separated cone chart.

Both placements and every finite lower differential term are retained with $z$ untested.

The remaining problem has changed, but it has not disappeared: compose the complete reconstructed sum with the grouped cross-lines on the characteristic domain. No internal Fourier identity supplies that missing product theorem.

Successor package

A successor receives:

1. the unchanged XIV-H package;

2. the finite retarded-pole module and its exact normalization;

3. the pole identity, uniform estimates, and reconstruction map;

4. the two complete $K_5$ amplitude maps with $z$ retained;

5. their exact finite lower-degree remainder and internal wavefront bounds;

6. the common precomposition sum and its canonical-domain value;

7. the precise missing I3 joint operator map;

8. the still closed I4–I6 and J3–J5 gates.

It receives no transverse degree of ${\mathcal{T}}_{\mathrm{form}}$, no cancellation theorem, and no permission to call either seventeen-dimensional count an excess. It receives no ${R_1}^{(1)}$.

The remaining chain is

Aiming at XV-B does not replace any arrow by a name.

Conclusion

An internal pole amplitude is not a cross-line prescription. A reconstructed tree sum is not its characteristic product. A frequency remainder is not the quantum remainder theorem. An undefined quantum coefficient is not an anomaly.

Declarations

scope. The new construction is an explicit finite distribution-valued internal amplitude module. It includes a proved pole identity, uniform annular estimates, reconstruction, finite lower-degree remainder, and internal wavefront bounds. No general intersecting-Lagrangian calculus is claimed.

common product phase. Separate Fourier transformations represent existing independent internal tensor factors. No sum of the retarded and quantum cross-line phases is formed or declared nondegenerate.

new prescription. The boundary value in the internal multiplier is fixed by the already declared retarded kernel. Its Fourier computation does not select a value for the absent characteristic graph product. No characteristic ambiguity is added to ${\mathfrak{C}}_{\mathrm{part}}$.

-variable scope. The variable $z$ is retained in the adapted coordinates and in the reconstructed kernels. Uniformity is joint distributional control on compact sets, not a pointwise restriction at a singular residual value.

scope. The finite lower-degree estimate concerns the internal normal-frequency module only. It controls every lower term of the finite $K_5$ differential formula, but not the later complete quantum graph.

and degree scope. The internal diagonal and flow-out intersection remains clean. It is not XIV-F's excluded matching. No internal conormal order or amplitude homogeneity is used as a transverse scaling degree of ${\mathcal{T}}_{\mathrm{form}}$.

scope. The analytic manuscript inputs are those listed in I0. XV-A is used only to state the exit prerequisite. XIV-A and XIV-B, the auxiliary value $-8I$, XIII-E's family, and resolved paired constructions supply no analytic map here.

scope. No ${R_1}^{(1)}$, interacting differential, Ward defect, quantum master-equation defect, anomaly, state, numerical loop amplitude, new vertex, or sharp-width limit is constructed.

. No successful pdflatex execution, proof-assistant verification, independently executed symbolic computation, independent referee review, or journal acceptance is asserted. The claimed mathematical results are the explicit proofs above.

XIVH

N. Cheung, Landmark XIV-H: Internal kernel relations on the operator route. Supplied manuscript XIV-H.tex. Immediate predecessor: exact internal maps, regular charts, transition geometry, and the H2 amplitude stop.

XIVG

N. Cheung, Landmark XIV-G: Operator and pullback composition of the regrouped retained germ. Supplied manuscript XIV-G.tex. Testing limitation, factorwise audit, and G2 boundary.

XIVF

N. Cheung, Landmark XIV-F: Paired composition on a proposed excess bundle. Supplied manuscript XIV-F.tex. Tangent-defect and projection geometry, recorded only as boundaries.

XIVE

N. Cheung, Landmark XIV-E: Joint symbol of the six-branch retained germ, or one named characteristic prescription. Supplied manuscript XIV-E.tex. Exact regrouping, complete root, tested-kernel bound, and characteristic lifts.

XIVD

N. Cheung, Landmark XIV-D: Cancellation or prescription on the retained noncoincident characteristic stratum. Supplied manuscript XIV-D.tex. Six-branch germ, actual local incidences, and branch-limited fiber numerator.

XIVC

N. Cheung, Landmark XIV-C: The joint two-line extension of one selected binary pair. Supplied manuscript XIV-C.tex. Retained characteristic stratum, positive relation, and contact rigidity.

XIIIC

N. Cheung, Landmark XIII-C: The joint two-line extension of one selected binary pair, on the unchanged carrier. Supplied manuscript XIII-C.tex. Four successor obligations and primitive collision extensions.

XIIIB

N. Cheung, Landmark XIII-B: One binary causal datum for the displayed coefficient, on the unchanged carrier. Supplied manuscript XIII-B.tex. Tree datum, retarded convention, ordered values, and properness boundary.

XIIE

N. Cheung, Landmark XII-E: Assembly of the second-order relational coefficient after admission of the density-weighted cubic insertion. Supplied manuscript XII-E.tex. Slot sign, roots, and existing completion kernels.

XIIC

N. Cheung, Landmark XII-C: One declared cubic weighting: actual Euler source, changed Noether identity, and the coupled-jet question. Supplied manuscript XII-C.tex. Fixed density representative and actual source.

XVA

N. Cheung, Landmark XV-A: The free quantum BV carrier for the ramp Einstein–scalar multiplet. Supplied manuscript XV-A.tex. Recorded free-carrier prerequisite and comparison boundary only; not an analytic input to the coefficient.

Hormander

L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, second edition, Springer, 1990. Fourier distributions, tensor products, pullbacks, and wavefront calculus.

Wunsch

J. Wunsch, Microlocal analysis and evolution equations: Lecture notes from the 2008 CMI/ETH Summer School, arXiv:0812.3181v3, 2023. Section 8: individual conormal and Lagrangian conventions. No general transition theorem is imported from these notes.

BGP

C. Bär, N. Ginoux, and F. Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007, arXiv:0806.1036. Causal Green operators and support conventions. The internal multiplier used here is computed explicitly.

FR

K. Fredenhagen and K. Rejzner, Perturbative algebraic quantum field theory, arXiv:1208.1428v2, 2013. Functional products and their domain distinctions. No renormalized product for the present nonlocal pair is imported.

BF

R. Brunetti and K. Fredenhagen, “Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds,” Communications in Mathematical Physics 208 (2000), 623–661, arXiv:math-ph/9903028. Recorded extension framework only; not applied to the missing characteristic target.

Dang

N. V. Dang, “The extension of distributions on manifolds, a microlocal approach,” Annales Henri Poincaré 17 (2016), 819–859, arXiv:1412.2808. Recorded microlocal extension framework only; no characteristic prescription is imported.